Backward Euler method

The implicit Euler method (after Leonhard Euler ) (also backward Euler method ) is a numerical method for solution of initial value problems. It is an implicit method, that is, in each step a - are solved equation - in general nonlinear.

The procedure

For the numerical solution of the initial value problem:

For an ordinary differential equation we choose a discretization, consider the discrete time points

And compute the iterated values

The value here is not given explicitly, but only implicitly. They are calculated so (depending on the type of right-hand side f) linear or non- linear system of equations must be solved in each iteration step.

The calculated values ​​are then approximations to the actual values ​​of the exact solution of the initial value problem dar. The smaller one chooses the step size, the more computational work has one, but the better the approximated values ​​.

If a procedure is defined above, we obtain the explicit Euler method.

Properties

The implicit Euler method has consistency and convergence of Procedure 1, it is A-stable, so its stability region contains the entire left half-plane of the complex plane. There is thus for the implicit Euler method no restrictions on the time steps due to stability limitations, which compensates for the compulsion of solving systems of equations in each step. Due to the low order, it is therefore of particular interest for problems in which the iteration runs into a stable final state, and the accuracy of the intermediate results is not interesting.